A306387 Number of partitions of sigma_1(n) into divisors of n.

1, 2, 2, 6, 2, 27, 2, 26, 7, 31, 2, 574, 2, 38, 33, 166, 2, 879, 2, 924, 39, 52, 2, 23732, 9, 59, 47, 1403, 2, 34256, 2, 1626, 55, 73, 47, 230819, 2, 80, 61, 50888, 2, 65638, 2, 2709, 1734, 94, 2, 2117920, 11, 3038, 77, 3536, 2, 113448, 65, 97298, 83, 115, 2, 19613170, 2, 122, 2601, 25510, 73, 180350

Offset

1,2

Comments

The equality sigma_1(n) = Sum{d|n} d defines one partition of sigma_1(n) into distinct divisors of n. This sequence gives the number of partitions of sigma_1(n) into not necessarily distinct divisors of n.

For prime number p, sigma_1(p) = p+1 and there are only two partitions: p and 1+1+1+...+1 (p summands).

Links

Table of n, a(n) for n=1..66.

Example

For n = 4, sigma_1(4) = 7, divisors(4) = {1,2,4} and 7 = 4+2+1 = 4+1+1+1 = 2+2+2+1 = 2+2+1+1+1 = 2+1+1+1+1+1 = 1+1+1+1+1+1+1.
For n = 9, sigma_1(9) = 13, divisors(9) = {1,3,9} and 13 = 9+3+1 = 9+1+1+1+1 = 3+3+3+3+1 = 3+3+3+1+1+1+1 = 3+3+1+1+1+1+1+1+1 = 3+1+1+1+1+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1+1+1.

Prog

(Magma) v:=[1..47];
for u in v do
u, #RestrictedPartitions(SumOfDivisors(u), {d:d in Divisors(u)});
end for;
(Magma)
a:= func< n | #RestrictedPartitions(SumOfDivisors(n), {d:d in Divisors(n)}) >; [ a(n) : n in [1..47] ];
(PARI) numbpartUsing(n, v, mx=#v)=if(n<1, return(n==0)); sum(i=1, mx, numbpartUsing(n-v[i], v, i)) \\ inefficient;
a(n) = numbpartUsing(sigma(n), divisors(n)); \\ after A018818; Michel Marcus, Feb 27 2019

Crossrefs

Cf, A000005, A000041, A000203, A018818.

Sequence in context: A286376 A100346 A359004 * A308692 A319352 A300834

Adjacent sequences: A306384 A306385 A306386 * A306388 A306389 A306390

Keyword

nonn

Author

Marius A. Burtea, Feb 26 2019

Status

approved